Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T13:51:30.927Z Has data issue: false hasContentIssue false

Contactless rebound of elastic bodies in a viscous incompressible fluid

Published online by Cambridge University Press:  24 May 2022

G. Gravina
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
S. Schwarzacher*
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
O. Souček
Affiliation:
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
K. Tůma
Affiliation:
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
*
Email address for correspondence: schwarz@karlin.mff.cuni.cz

Abstract

In this paper, we investigate the phenomenon of particle rebound in a viscous incompressible fluid environment. We focus on the important case of no-slip boundary conditions, where it is now well established that collisions cannot occur in finite time under certain assumptions. In a simplified framework, we provide conditions which allow us to prove that rebound is possible even in the absence of a topological contact. Our results lead to the conjecture that a qualitative change in the shape of the solid is necessary to obtain a physically meaningful rebound in fluids. We support the conjecture by comparing numerical simulations performed for the reduced model with finite element solutions obtained for corresponding well-established partial differential equation systems describing elastic solids interacting with incompressible fluids.

Type
JFM Papers
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E. & Wells, G.N. 2015 The FEniCS project version 1.5. Arch. Numer. Softw. 3 (100).Google Scholar
Barnocky, G. & Davis, R.H. 1989 The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J. Fluid Mech. 209, 501519.CrossRefGoogle Scholar
Bassani, R. & Piccigallo, B. 1992 Hydrostatic Lubrication. Tribology Series, vol. 22. Elsevier Science.Google Scholar
Benesova, B., Kampschulte, M. & Schwarzacher, S. 2020 A variational approach to hyperbolic evolutions and fluid–structure interactions. arXiv:2008.04796.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3), 242251.CrossRefGoogle Scholar
Casanova, J.-J., Grandmont, C. & Hillairet, M. 2021 On an existence theory for a fluid-beam problem encompassing possible contacts. J. Éc. polytech. Math. 8, 933971.CrossRefGoogle Scholar
Cooley, M.D.A. & O'Neill, M.E. 1969 On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere. Mathematika 16 (1), 3749.CrossRefGoogle Scholar
Cox, R.G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface-II small gap widths, including inertial effects. Chem. Engng Sci. 22 (12), 17531777.CrossRefGoogle Scholar
Davis, R.H., Serayssol, J.-M. & Hinch, E.J. 1986 The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163, 479497.CrossRefGoogle Scholar
Donea, J., Huerta, A., Ponthot, J.-P. & Rodríguez-Ferran, A. 2004 Arbitrary Lagrangian–Eulerian Methods, chap. 14. American Cancer Society.CrossRefGoogle Scholar
Dunne, T. & Rannacher, R. 2006 Adaptive finite element approximation of fluid–structure interaction based on an Eulerian variational formulation. In Fluid–Structure Interaction (ed. H.-J. Bungartz & M. Schäfer), pp. 110–145. Springer.CrossRefGoogle Scholar
Feireisl, E. 2003 On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167, 281308.CrossRefGoogle Scholar
Feireisl, E. 2004 On the Motion Of Rigid Bodies in a Viscous Incompressible Fluid, pp. 419441. Birkhäuser.Google Scholar
Gérard-Varet, D. & Hillairet, M. 2010 Regularity issues in the problem of fluid structure interaction. Arch. Ration. Mech. Anal. 195 (2), 375407.CrossRefGoogle Scholar
Gérard-Varet, D. & Hillairet, M. 2014 Existence of weak solutions up to collision for viscous fluid–solid systems with slip. Commun. Pure Appl. Maths 67 (12), 20222075.CrossRefGoogle Scholar
Gérard-Varet, D., Hillairet, M. & Wang, C. 2015 The influence of boundary conditions on the contact problem in a 3D Navier–Stokes flow. J. Math. Pure Appl. 103 (1), 138.CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W., Hesla, T.I. & Joseph, D.D. 1999 A distributed lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 25 (5), 755794.CrossRefGoogle Scholar
Grandmont, C. 2002 Existence for a three-dimensional steady state fluid–structure interaction problem. J. Math. Fluid Mech. 4 (1), 7694.CrossRefGoogle Scholar
Grandmont, C. & Hillairet, M. 2016 Existence of global strong solutions to a beam-fluid interaction system. Arch. Ration. Mech. Anal. 220, 12831333.CrossRefGoogle Scholar
Hagemeier, T., Thévenin, D. & Richter, T. 2021 Settling of spherical particles in the transitional regime. Int. J. Multiph. Flow 138, 103589.CrossRefGoogle Scholar
Hesla, T.I. 2004 Collisions of smooth bodies in viscous fluids: a mathematical investigation. PhD thesis, University of Minnesota.Google Scholar
Hillairet, M. 2007 Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Part. Diff. Equ. 32 (7–9), 13451371.CrossRefGoogle Scholar
Hillairet, M., Seck, D. & Sokhna, L. 2018 Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp. C. R. Math. Acad. Sci. Paris 356 (11–12), 11561166.CrossRefGoogle Scholar
Hillairet, M. & Takahashi, T. 2009 Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal. 40 (6), 24512477.CrossRefGoogle Scholar
Janela, J., Lefebvre, A. & Maury, B. 2005 A penalty method for the simulation of fluid – rigid body interaction. ESAIM: Proc. 14, 115123.CrossRefGoogle Scholar
Joseph, G.G. 2003 Collisional dynamics of macroscopic particles in a viscous fluid. PhD thesis, California Institute of Technology.Google Scholar
Joseph, G.G., Zenit, R., Hunt, M.L. & Rosenwinkel, A.M. 2001 Particle–wall collisions in a viscous fluid. J. Fluid Mech. 433, 329346.CrossRefGoogle Scholar
Leal, L.G. 1992 Chapter 7 – thin films, lubrication, and related problems. In Laminar Flow and Convective Transport Processes (ed. L. Gary Leal), pp. 345–448. Butterworth-Heinemann.CrossRefGoogle Scholar
Liu, C. & Walkington, N.J. 2001 An Eulerian description of fluids containing visco-elastic particles. Arch. Ration. Mech. Anal. 159 (3), 229252.CrossRefGoogle Scholar
Olsson, E. & Kreiss, G. 2005 A conservative level set method for two phase flow. J. Comput. Phys. 210, 225246.CrossRefGoogle Scholar
Richter, T. 2017 Fluid–Structure Interactions: Models, Analysis and Finite Elements, vol. 118. Springer.CrossRefGoogle Scholar
Scovazzi, G. & Hughes, T. 2007 Lecture notes on continuum mechanics on arbitrary moving domains. Tech. Rep. SAND-2007-6312P. Sandia National Laboratories.Google Scholar
Starovoitov, V.N. 2004 Behavior of a rigid body in an incompressible viscous fluid near a boundary. In Free boundary problems (Trento, 2002), Internat. Ser. Numer. Math., vol. 147, pp. 313–327. Birkhäuser.CrossRefGoogle Scholar
Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. & Matsumoto, Y. 2011 A full Eulerian finite difference approach for solving fluid–structure coupling problems. J. Comput. Phys. 230 (3), 596627.CrossRefGoogle Scholar
Vázquez, J.L. & Zuazua, E. 2006 Lack of collision in a simplified 1D model for fluid–solid interaction. Math. Models Meth. Appl. Sci. 16 (5), 637678.CrossRefGoogle Scholar
von Wahl, H., Richter, T., Frei, S. & Hagemeier, T. 2020 Falling balls in a viscous fluid with contact: comparing numerical simulations with experimental data. Preprint arXiv:2011.08691.CrossRefGoogle Scholar